Is the Inverse Element in Modular Arithmetic Unique?

[alphamu]

Question: Let n be a positive integer and consider x,y,z to be elements of Zn. Prove that if x . y = 1 and x . z = 1, then y = z. (Since working in Zn the sign '.' means "multiplication modulo n".)

Conclude that if x has an inverse element in Zn, then the inverse element is unique.

Attempt: Well if x . y = 1 then this means that xy = n + 1 since were working in Zn. This means the same can be said about y . z = 1 that yz = n + 1. This suggests that z = x by substitution. I'm not sure of this is correct or if I'm going down right path here.

Other thing I thought about was multiplying x . y = 1 by z on both sides to prove y=z but I'm a bit stuck.

I'm also a bit confused about the last bit of the question where I have to conclude if x has inverse element (not entirely sure what this is) in Zn then the inverse element is unique.

Can anyone help please?Sent from my iPhone using Physics Forums

 

[HallsofIvy]

Question: Let n be a positive integer and consider x,y,z to be elements of Zn. Prove that if x . y = 1 and x . z = 1, then y = z. (Since working in Zn the sign '.' means "multiplication modulo n".)

Conclude that if x has an inverse element in Zn, then the inverse element is unique.

Attempt: Well if x . y = 1 then this means that xy = n + 1 since were working in Zn. This means the same can be said about y . z = 1 that yz = n + 1.

This is NOT true. for example, 5*5= 1 (mod 12) because 5*5= 2(12+ 1). That is, 2n+ 1, NOT n+ 1.

This suggests that z = x by substitution. I'm not sure of this is correct or if I'm going down right path here.

Other thing I thought about was multiplying x . y = 1 by z on both sides to prove y=z but I'm a bit stuck.

Why "a bit stuck"? Where did you get stuck? Since multiplication is commutative, yes, if xy= 1 then zxy= (xz)y= z so y= z.

I'm also a bit confused about the last bit of the question where I have to conclude if x has inverse element (not entirely sure what this is) in Zn then the inverse element is unique.

An "inverse element" of x is another element, y, such that xy= yx= 1. For any "other inverse element", z, xz= 1.

Can anyone help please?


Sent from my iPhone using Physics Forums

I'm a bit puzzled why, on recognizing that you did not know the definitions well ("not entirely sure what this is"), you did not immediately look up and review the definitions.